#433824

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6,8,10)(7,9,11,12)(13,14)(15,25)(16,30)(17,20)(18,24)(19,23)(21,28)(22,32)(26,40)(27,37)(29,34)(31,38)(33,35)(36,41)(39,42)(43,44);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(14,20)(16,31)(21,34)(23,24)(25,37)(26,41)(32,33)(42,43);(7,11)(9,12)(13,15,19,29,17,27,18,28)(14,21,24,37,20,34,23,25)(16,26,33,43,31,41,32,42)(22,36,38,44,35,40,30,39);(2,4)(13,16)(14,22)(15,26)(17,31)(18,32)(19,33)(20,35)(21,36)(23,30)(24,38)(25,39)(27,41)(28,42)(29,43)(34,40)(37,44)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341020);